Question

Instruction. Show all your work for full credit. Partial credit will be awarded for a substantial progress on a problem. 1. (12 points) Let R be the region bounded by the curves $y = x^2$ and $y = 3x$. Calculate the following quantities. 10f Y (3,9) a) The area of R. Answer b) The volume obtained by rotating R about the x-axis. Answer c) The volume obtained by rotating R about the y-axis. Answer 2. (4 points) The engine of a bus exerts a force $F(x) = 3000 - 2x$ pounds on the bus when it has gone x feet from its starting point. How much work does the engine do on the bus in the first 1000 feet? d$w = F \cdot dx$ $\int_{0}^{1000} dw = \int_{0}^{1000} (3000 - 2x) dx$ $w = [3000x - \frac{2x^2}{2}]_{0}^{1000}$ $= [(3000 \times 1000 - (1000)^2) - (0 - 0)]$ $w = 2 \times 10^6 lb \cdot ft$ Answer $2 \times 10^6 lb \cdot ft$ 3. (4 points) Records indicate that t hours past midnight, the temperature at the local airport was $P(t) = -0.1t^2 + t + 50$ degrees Fahrenheit. What was the average temperature at the airport between 9 AM and noon? Answer 1

          Instruction. Show all your work for full credit. Partial credit will be awarded for a substantial
progress on a problem.
1.
(12 points) Let R be the region bounded by the curves $y = x^2$ and $y = 3x$. Calculate the
following quantities.
10f Y
(3,9)
a) The area of R.
Answer
b) The volume obtained by rotating R about the x-axis.
Answer
c) The volume obtained by rotating R about the y-axis.
Answer
2.
(4 points) The engine of a bus exerts a force $F(x) = 3000 - 2x$ pounds on the bus when
it has gone x feet from its starting point. How much work does the engine do on the bus in the
first 1000 feet?
d$w = F \cdot dx$
$\int_{0}^{1000} dw = \int_{0}^{1000} (3000 - 2x) dx$
$w = [3000x - \frac{2x^2}{2}]_{0}^{1000}$
$= [(3000 \times 1000 - (1000)^2) - (0 - 0)]$
$w = 2 \times 10^6 lb \cdot ft$
Answer
$2 \times 10^6 lb \cdot ft$
3.
(4 points) Records indicate that t hours past midnight, the temperature at the local airport
was
$P(t) = -0.1t^2 + t + 50$
degrees Fahrenheit. What was the average temperature at the airport between 9 AM and noon?
Answer
1

Instruction. Show all your work for full credit. Partial credit will be awarded for a substantial
progress on a problem.
1.
(12 points) Let R be the region bounded by the curves y = x^2 and y = 3x. Calculate the
following quantities.
10f Y
(3,9)
a) The area of R.
Answer
b) The volume obtained by rotating R about the x-axis.
Answer
c) The volume obtained by rotating R about the y-axis.
Answer
2.
(4 points) The engine of a bus exerts a force F(x) = 3000 - 2x pounds on the bus when
it has gone x feet from its starting point. How much work does the engine do on the bus in the
first 1000 feet?
dw = F · dx
∫0^1000 dw = ∫0^1000 (3000 - 2x) dx
w = [3000x - (2x^2)/(2)]0^1000
= [(3000 × 1000 - (1000)^2) - (0 - 0)]
w = 2 × 10^6 lb · ft
Answer
2 × 10^6 lb · ft
3.
(4 points) Records indicate that t hours past midnight, the temperature at the local airport
was
P(t) = -0.1t^2 + t + 50
degrees Fahrenheit. What was the average temperature at the airport between 9 AM and noon?
Answer
1

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